II PUC mathematics imp qns

8/7/2019 II PUC mathematics imp qns

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BY SUCHINDRA D &KHV for http://pucpcmb.wordpress.com1

PREPARING FOR II PUC-ANNUAL EXAMINATION: MATHEMATICS

QUESTIONS ONE SHOULD NOT NEGLECT

NUMBER THEORY

1. Find the GCD of the following using Euclid Algorithm, represent them as a linear pair of itand prove that the expression is not unique.a) 506 and 1155 b) 495 and 675 c) 216 and 6125 d) 408 and 1032 c) 275 and 726

2. Find the total number of all positive divisors and sum of all positive divisors of thefollowing.a) 39744 b) 756 c) 432 d) 960 e) 252

f ) 2010 g) &%$ h) 352 i) 891 j) 720

3. Prove the following :

a) Divisibility is a transitive relation.b) If I are relatively prime and I | , then I | b .

c) If p is a prime number and p | ab , then p | a or p | b .d) The relation ‘ congruence modulo n’ is an equivalence relation.e) If {J 2 is a positive divisor of m, then {J.f) If {J and is any integer then { - { -

{J .

g) If I I{J and {I then prove that {J .

MATRICES AND DETERMINANTS

1. Solve by matrix method and Cramer’s rule :a) . - - . - . . i) - . b) - . . - - . j) - - . c) - . - . - - . k) . . d) . - - - - l) - . e) . . - - . . 2.

2. State and verify Cayley Hamilton theorem.

a) b) . c) d) . . e) . 3. State Cayley Hamilton theorem and hence find the inverse of the following

a) . . b) . c) . . d) . e) 4. Prove the following

a) $ $ I I$ { . { . I{I . b) $ - I $ - I I$ - I .{ . { . I{I . c) - - I I - I - I I - - { - - I%




d) . . I I . I . I I . . { - - I%

e) - I - I - I

${ - - - I f) $ $ $

g) ~ $ - $ - $ - ~ { . { . { . { - -

h ) $ - I$ $ $$ I$ - $ $I$ I$ $ - $ $$I$

i)

I $I $ I I$ { . { . I{I . { - I - I

ii)

j) $ I I - I$$ - $ I $ - I I$ $$I$

k) $ - I $ - II I I$ - - $ - $ - I$

l) - - I .I ..I - - I .. . - - I { - { - I{I - VECTORS

1. Prove the following using vectors :a) Medians of a triangle are concurrent. b)The angle in a semicircle is a right anglec). Sine rule d). Cosine rule e). Projection rulef). ��{ - ��{ . ��{ - ��{ .


a). ? - z z - II - C ?z IC b) ? . z z . II . C c).z 0 I

d) 0 { 0 - 0 { 0 - 0 0 f). ? 0 z z 0 II 0 C ?z IC$

g). ?z z - I - z - IC

3. Find the volume of parallelepiped whose co-terminal edges are :

a) - - . - - - b). . . . - -

4. If the following vectors are coplanar find m :

a) . - - . - - . b) . - - . . -




5. If the vectors . - - - - - - - are position vectors of the four

coplanar points, find .6. If the vectors - . . - are orthogonal, find .

7. Find the cosine and sin of the angle between the vectors :

a. . - . . b. . -

GROUPS

1 . Define : Group , algebraic structure, Semigroup, Monoid, Abelian group, Subgroup.


1. A non empty subset H of a group G is a subgroup of G iff H # H.2. A non empty subset H of a group G is a subgroup of G iff H satisfies closure law and inverse law.3. Intersection of two subgroups is also a subgroup. 4. Identity element of a group is unique.5. inverse of an element is unique. 6. {## .7. A group of order 3 is Abelian.

8. Cancelation laws hold is a group.9. There exist a unique solution for the equation .10. If $ , then prove G is Abelian.11. If {$ $$ , then prove that G is abelian.

12. If every element of a group is its own inverse OR $ prove that G is Abelian.

3. Prove the following is an Abelian group :

1. On Z define - - 2. On define $%

3. On . . define - - 5. On

. define

- .

6. Set of all integral powers of 5 under multiplication.7. Set of all complex numbers whose modulus is unity under multiplication.8. Set of all 0 matrices under addition.

9. under multiplication

10. Cube roots of unity under multiplication11. Fourth roots of unity under multiplication

12. under #" 13. under #" 14.

CIRCLES1 . Prove the following :

1 . Define orthogonal circles and derive the condition for two circles to be orthogonal.2. Derive the condition for - I to be a tangent to the circle $ - $ $, and hence find the

point of contact. 3.. Find the equation of tangent to the circle $ - $ - - - I at{# # on it.4. Find the length of tangent to the circle $ - $ - - - I from {# #.5. Define radical axis of two circles, derive the equation of radical axis of two circles and prove thatradical axis is perpendicular to line joining the centers.

6. Find the midpoint of the chord - of the circle $ - $ .7. Find the equation of the common chord of $ - $ - . - $ - $ - - . .

8. Find the radical axis OR equation of common chord OR equation of common tangent of the circles :

1.

$ - $ - - &

$ - $ . . - .

2. $ - $ . - - $ - $ . - -

3. $ - $ - - . $ - $ - - - .




9. Find the length of the common chord of the circles :

i) {$ - $ . . - $ - $ - . -

ii) $ - $ - . - $ - $ - - -

iii) $ - $ - - - $ - $ - - -

10. Find the equation of the circle passing through

{with its centre on

- . and cutting

orthogonally $ - $ . - .

CONIC SECTIONS

1 . Define ellipse, derive the equation of ellipse in standard form.

2. Define hyperbola, derive the equation of hyperbola in standard form.

3. Define parabola, derive the equation of parabola in standard form.

TRIGONOMETRY

GENERAL SOLUTION OF GENERAL SOLUTION OF GENERAL SOLUTION OF GENERAL SOLUTION OF TRIGONOMETRIC EQUATRIGONOMETRIC EQUATRIGONOMETRIC EQUATRIGONOMETRIC EQUATIONSFind the general solution/s for the following:

1. 03sinsin8 3 =− xx 2. 0cos83cos 3 =+ θ θ

3. 1)sin(cos2 44 =+ xx 4. 06sin4sin2sin =++ xxx

5. 03cos2coscos =++ xxx 6. xxx 3tan2tantan =+

7. xxxx cos2cos4cos5cos −=− 8. xxxx 3sin2cos3cos2sin −=−

9. xxxx 2cos6sincos5sin = 10. θ θ θ θ 2sin4cos6sin8cos =

11. xxcoxxx 35cos7cos9cos = 12. xxxx sin3sin5sin7sin =

13. 13cos2coscos4 =xxx 14. 22sec4sec =− θ θ

15. 2sin)13(cos)13( =−++ θ θ 16. 3sectan =+ xx

17. 3cos2cot =+ ecxx 18.°=°=+

=++

15cot75tan32:int

1sincos)32(

H

θ θ

19. θ θ θ sincos2cos += 20. 2sin)13(cos)13( =++− xx

21. 01sectan3 =+++ xx 22. 1sec2tan3 −= xx

23. 2sincos −=+ xx 24. 1cossin2sincos22

=−− θ θ θ θ




INVERSE TRIGONOMETRIC FUNCTIONS:INVERSE TRIGONOMETRIC FUNCTIONS:INVERSE TRIGONOMETRIC FUNCTIONS:INVERSE TRIGONOMETRIC FUNCTIONS:

1. Prove that:

i)

65

33cos

13

5sin

5

4cos

111 −−−=+ ii)

65

33cos

13

12cos

5

3sin

111 −−−=+

iii)25

3sin

2

1tan2

11 π =+

−−iv) 3cot

18

1tan

8

1tan

7

1tan

1111 −−−−=++

v)27

9tan

7

4tan

7

1tan

111 π =++

−−−vi)

48

1tan

7

1tan

5

1tan

3

1tan

1111 π =+++

−−−−

2. Prove that:

i)a

b

b

a

b

a 2cos

2

1

4tancos

2

1

4tan 11 =

−+

+ −− π π

ii)21

22

22

cos2

1

411

11tan x

xx

xx −+=

−−+

−++ π

3. If π =++ −−−z yx

111 tantantan , then prove that x+y+z=xyz.

4. If π =++ −−−

cba

1cot

1cot

1cot 111

, prove that a+b+c=abc.

5. If 2

tantantan 111π

=++ −−− z yx , prove that xy+yz+zx=1.

6. If π =++ −−−z yx

111 coscoscos , prove that x2+y2+z2+2xyz=1.

7. If 2

sinsinsin 111 π =++ −−− z yx , prove that x2+y2+z2+2xyz=1.

8. If π =++ −−−z yx

111 sinsinsin , prove that z xyyx =−+− 22 11 .

9. Solve for ‘x’:

i)2

)1(tan)1(tan 11 π =−++ −− xx ii) xxxx 3tan)1(tan2tan)1(tan 1111 −−−− =+++−

iii)42

1tan

2

1tan 11 π

=+

++

−

− −−

x

x

x

xv)

3

2

2

1cot

1

2tan

21

2

1 π =

−+

−

−− x

x

x

ix) 3cot2tan11 π

=+−−

xx x) 3

22sinsin

11 π

=+−−

xx

II PUC mathematics imp qns

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